harlene tosses two number cubes. if a sum of 8 or 12 comes up, she gets 9 points. if not, she loses 2…

harlene tosses two number cubes. if a sum of 8 or 12 comes up, she gets 9 points. if not, she loses 2 points. what is the expected value of the number of points for one roll?\n$-\frac{2}{3}$\n$-\frac{1}{6}$\n$\frac{1}{6}$\n$\frac{2}{3}$

harlene tosses two number cubes. if a sum of 8 or 12 comes up, she gets 9 points. if not, she loses 2 points. what is the expected value of the number of points for one roll?\n$-\frac{2}{3}$\n$-\frac{1}{6}$\n$\frac{1}{6}$\n$\frac{2}{3}$

Answer

Explanation:

Step1: Find total outcomes

When tossing two number cubes (each with 6 faces), total outcomes = (6\times6 = 36).

Step2: Find favorable outcomes for sum 8 or 12

  • Sum of 8: Possible pairs ((2,6),(3,5),(4,4),(5,3),(6,2)) → 5 outcomes.
  • Sum of 12: Only ((6,6)) → 1 outcome.
  • Total favorable for 8 or 12: (5 + 1=6) outcomes.

Step3: Calculate probabilities

  • Probability of sum 8 or 12 ((P(\text{win}))): (\frac{6}{36}=\frac{1}{6}).
  • Probability of not getting 8 or 12 ((P(\text{lose}))): (1-\frac{1}{6}=\frac{5}{6}).

Step4: Calculate expected value

Expected value ((E)) = (Points for win (\times) (P(\text{win}))) + (Points for lose (\times) (P(\text{lose})))
(E = 9\times\frac{1}{6}+(-2)\times\frac{5}{6})
(E=\frac{9}{6}-\frac{10}{6}=\frac{9 - 10}{6}=-\frac{1}{6}).

Answer: (-\frac{1}{6}) (corresponding to the option with (-\frac{1}{6}))